Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}6x+3y &= 4 \\ 2x+6y &= -7\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-2$ and the bottom equation by $1$ $\begin{align*}-12x-6y &= -8\\ 2x+6y &= -7\end{align*}$ Add the top and bottom equations. $-10x = -15$ Divide both sides by $-10$ and reduce as necessary. $x = \dfrac{3}{2}$ Substitute $\dfrac{3}{2}$ for $x$ in the top equation. $6( \dfrac{3}{2})+3y = 4$ $9+3y = 4$ $3y = -5$ $y = -\dfrac{5}{3}$ The solution is $\enspace x = \dfrac{3}{2}, \enspace y = -\dfrac{5}{3}$.